I can find sources on the internet that provides expression of the derivative of $Q$ and $R$ with respect to $A$ in the expression $A=QR$. One such example is https://j-towns.github.io/papers/qr-derivative.pdf. However, I have not yet found a proof on whether $QR$ factorization of $A$ is continous in $A$. I guess this must be the case, since people take the time to diffrentiate it, and that wikipedia claim that it is numerically stable.
More concretely:
Let $(A_k)_{k=1}^{\infty}$ be a series of real $m\times n$ matrices. By QR decomposition, define $Q_k$ and $R_k$ by $A_k=Q_kR_k$. Let $\lim_{k\to \infty}A_k = A$ for some real $m\times n$ matrix $A$. Let $A=QR$ by QR factorization.
Is it so that $\lim_{k\to \infty}Q_k = Q$ and $\lim_{k\to \infty}R_k = R$? Why? Does it matter whether all the matrices $A_k$ and $A$ has the same rank or not?